Below we demonstrate this crucial point more clearly via providing an alternative formulation of
finite remainders introduced in the proposed prescription, which also helps to clarify a few points
alluded in the preceding section.
Let us consider the finite remainders of amplitudes in CDR as defined by the celebrated amplitude-level
factorization formula.
Singularities in the dimensionally regularized QCD amplitudes are known to
factorize [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}.
For our purpose, we can sketch this factorization property
of a bare QCD scattering amplitude \(\hat{\mathcal {A}}(\epsilon )\)  among several resolved
external particles (with fixed external kinematics) schematically as follows:
\(\hat{\mathcal {A}}(\epsilon ) = \hat{\mathcal {Z}}_{\mathrm {IR}}(\epsilon )~\mathcal {Z}_{\mathrm {UV}}(\epsilon )~ \hat{\mathcal {F}}(\epsilon ) \, ,\) 
